= Solution
{c}
The <Hiemenz flow> profile changes on $\eta=O(1)$, so its <boundary layer thickness> is
$$
\boxed{\delta_{\mathrm{BL}}\sim\sqrt{\frac\nu A}.}
$$
A specified edge convention, such as $u/(Ax)=0.99$, introduces an order-one numerical multiplier requiring the profile to be solved. Scaling alone does not determine that multiplier. The result has no separate dependence on $x$, $y$, $\rho$ or $p_0$ at fixed <kinematic viscosity>. At fixed <dynamic viscosity>, it is $\sqrt{\eta/(\rho A)}$.
The <viscous diffusion time> across a thickness $\delta$ is $\delta^2/\nu$; the imposed strain has time scale $A^{-1}$. Their balance sets the thickness. Increasing <kinematic viscosity> spreads the layer, while increasing the imposed <velocity gradient> presses the layer toward the wall. The outward tangential <velocity> increases as $Ax$, so the decreasing travel time per unit distance offsets the usual downstream broadening.
For a conventional estimate of transition to <turbulence>, form the local horizontal <Reynolds number>:
$$
\mathrm{Re}_x=\frac{(Ax)x}{\nu}=\frac{Ax^2}{\nu}.
$$
If transition is assumed to occur at an empirical value $\mathrm{Re}_{\mathrm{crit}}$, then
$$
\boxed{l\sim\sqrt{\frac{\mathrm{Re}_{\mathrm{crit}}\nu}{A}}.}
$$
A rough borrowed threshold of order $10^5$–$10^6$ would give $l$ of order $10^2$–$10^3\sqrt{\nu/A}$. This is a heuristic estimate, not a universal stability prediction: the accelerating outer flow and favorable <pressure gradient> can delay transition, and disturbance amplitudes and wall roughness matter. Simply setting $\mathrm{Re}_x\sim1$ locates failure of the thin-layer approximation near the <stagnation point>, not onset of <turbulence>.
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